It doesn’t really answer your question, but I came across this description earlier. I thought it was interesting that they made a bunch at once, and then just picked one of them to be the international standard, with the rest becoming national standards:
But for most objects, buoyancy is negligible, and in any event nobody said that we were in different atmospheres on the Earth and the Moon. You could do your measurements in a vacuum chamber on Earth, or in a sealed habitat on the Moon, and the balance scales wouldn’t care whether you were on the Earth or Moon. Any spring-type scale, though, would give very different readings on the Earth and Moon.